Evaluation of Production Performance from a Hydraulically Fractured Well

ABSTRACT

An analytical solution is obtained for a pseudo-steady state production from a vertically fractured well with finite or infinite fracture conductivity. The analytical solution may be used to compute a pseudo-steady state constant for the reservoir. Subsequently, performance parameters relating to the reservoir may be derived from the pseudo-steady state constant. For example, parameters such as production decline rate, total hydrocarbon reserves, and economically recoverable reserves for the reservoir may be computed. The disclosed analytical solution, instead of a conventional numerical simulation, can significantly speed up analysis and improve the accuracy of the calculation of these parameters.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 62/268,958 to Kangping Chen, filed on Dec. 17, 2015, andentitled “Evaluation Of Production Performance From A HydraulicallyFractured Well,” which is hereby incorporated by reference in itsentirety.

FIELD OF THE DISCLOSURE

The instant disclosure relates to extraction of underground resources.More specifically, this disclosure relates to determining performancefactors relating to the extraction of underground resources from aparticular well.

BACKGROUND

Production of hydrocarbon from a well is normally conducted with aconstant production rate over long periods, although the rate can bechanged during the productive life of the well due to maintenance andother technical requirements. FIG. 1 is a graph illustrating reservoirpressure change with time for a well producing at constant rate in aclosed reservoir. P_(i) is a reservoir initial pressure; P_(w) is awellbore pressure; and P_(cri) is the lowest permissible wellborepressure (critical pressure). The time sequence of the graph of FIG. 1is t₁<t₂<t₃<t₄<t₅<t₆ . . . . At the start of a production, reservoirpressure initially depletes in the immediate neighborhood of thewellbore, and this pressure drawdown spreads outward diffusively towardsthe reservoir outer boundary (as shown in FIG. 1). For a closed (sealed)reservoir, the no-flow reservoir boundary starts to affect the pressurewhen the spreading pressure depletion front approaches the boundary.When the boundary effect has been fully reflected in the pressure field,the spatial distribution of the pressure no longer changes with time andthe fluid flow reaches the so-called pseudo-steady state (lines 102A inFIG. 1). The flow prior to the pseudo-steady state flow is called thetransient flow (lines 102B in FIG. 1), the duration of which depends onhow fast the pressure drawdown diffuses in the reservoir, which in turnis determined by the reservoir and fluid properties, namelypermeability, porosity, viscosity and compressibility. For conventionalreservoirs where the permeability is greater than 0.1 mD (mini-Darcy),the transient flow period usually lasts from days to months; while forunconventional reservoirs which have permeabilities less than 0.1 mD,the period can last from years to even tens of years. For closedreservoirs, the pseudo-steady state flow is a dominant, long-durationand most productive flow regime, especially for conventional reservoirs.During the pseudo-steady state flow period, the wellbore bottom-holeflowing pressure (BHFP) decreases linearly in time in order to maintainthe constant production rate. However, once the bottom-hole flowingpressure has declined to the lowest permissible value, which is oftendetermined by the surface equipment limitations, a constant rateproduction can no longer be continued, and a constant pressureproduction must follow. The production rate for this constant pressureproduction period declines in time, eventually apporaching zero as thereservoir pressure approaches the lowest permissible wellbore pressure(lines 102C in FIG. 1).

Pseudo-steady state flow is a dominant flow regime during constant rateproduction from a finite, closed reservoir. For a verticallyfractured-well in a finite reservoir approximated as having an slightlyelliptical shape, conventional solutions exist for analyticallydetermining the flow for the case of infinite fracture conductivity. Forfinite fracture conductivity, conventional computational techniques toachieve a pseudo-steady state solution involve running numericalsimulations over long times of hours, days, or longer.

Pseudo-steady state flow is a dominant flow regime during constant rateproduction from a closed reservoir: after the effects of the no-flowcondition on the reservoir outer boundary have been fully reflected inthe flow field and the transients associated with the flow startup havedecayed to be negligible, the flow in the reservoir reaches a state inwhich the spatial distribution of the pressure no longer changes withtime. Pseudo-steady state flow is thus a boundary-dominated flow. Onedefinition for pseudo-steady state is the condition in a finite, closedreservoir when producing at a constant rate that “every point within thereservoir will eventually experience a constant rate of pressuredecline.” This constant rate of pressure decline is the result of massconservation for constant rate production from a closed reservoir. Thiscondition is sometimes referred to as pseudo-steady, quasi-steady,semi-steady, or even steady state. The term pseudo-steady is used herein reference to this particular flow regime.

Pseudo-steady state (PSS) can be a prolonged period of constant rateproduction from a closed reservoir. During this period, the reservoirpressure declines linearly with time, the rate of which is determined bythe specified production rate and the drainage area. The pseudo-steadystate solution provides the reservoir pressure distribution as well asthe productivity index for this important flow period. Once the bottomhole flowing pressure has declined to the lowest permissible value,however, a constant rate production can no longer be continued, and aconstant pressure production must follow. The production rate for thislatter constant pressure production period declines in time. Productionrate decline analysis for this period plays an important role forestimating the hydrocarbon reserves in place and for assessing theeconomically recoverable amount of fluid from a reservoir. Becausepseudo-steady state is the flow regime immediate preceding theproduction rate decline period, the pseudo-steady state solution hasbeen conventionally used in the production rate decline analysis forunfractured wells and for fractured wells. In these analyses, thepseudo-steady state dimensionless pressure drawdown at the wellbore isexpressed as

Δp _(wD,PSS)=2πt _(DA) +b _(D,PSS),   (1)

where t_(DA) is the drainage area based dimensionless time, andb_(D,PSS) is the so-called pseudo-steady state constant which depends onthe reservoir model as well as the well/reservoir configuration. Thispseudo-steady state constant b_(D,PSS) is used to define the appropriatedimensionless decline rate and time in many of the currently usedproduction decline rate analysis models. Furthermore, the pseudo-steadystate constant is the reciprocal of the dimensionless productivity indexJ_(D,PSS) for the pseudo-steady state, J_(D,PSS)=1/b_(D,PSS), whichmeasures the productivity of the well for this flow period. J_(D,PSS) isalso important for production optimization for a fractured well. Forunfractured wells, the pseudo-steady state constant b_(D,PSS) can beobtained analytically for reservoirs of very simple shapes. These exactanalytical solutions have been modified by shape factors and used asapproximate analytical solutions for other reservoir geometries. Forhydraulically fractured wells, however, exact analytical solution forb_(D,PSS) is not available. For a vertically fractured well withinfinite fracture conductivity, an exact analytical solution for thepseudo-steady state flow in a reservoir bounded by an ellipticalboundary is known, which leads to an analytical expression for thepseudo-steady state constant b_(D,PSS). For the more practical case offinite fracture conductivity, however, no exact analytical solution inthe physical variable space has been reported in the literature forpseudo-steady state flow. For finite fracture conductivity, oneconventional numerical procedure is to extract b_(D,PSS) by subtracting2πt_(DA) from the long-time numerical solution for constant rateproduction from a fractured well in an elliptical reservoir. Thisprocedure is quite time consuming; and curve-fitting has been used toobtain an empirical relation between b_(D,PSS) and the reservoirgeometric parameter and the fracture conductivity.

SUMMARY

An analytical solution for pseudo-steady state flow for a verticallyfractured well with finite fracture conductivity in a closed reservoirmodeled as having a nearly circular, slightly elliptical shape isdescribed in embodiments of the present invention. This analyticalsolution provides a solution to a problem with no previous knownanalytical solution. Furthermore, the analytical solution can be used incomputer simulations to improve production performance of ahydraulically fractured well, provide prospectors with improvedinformation for deciding on production wells, and improve productionfrom those wells selected for production. The analytical solution allowscomputer modeling to be performed accurately and timely. Conventionaltechniques described above failed to provide an analytical solution forpseudo-steady state flow for vertically fractured wells, and thoseconventional techniques consumed significant amounts of computerprocessing time.

The analytical solution can be expressed in terms of elementaryfunctions and provides a simple expression for the pseudo-steady stateconstant and the dimensionless productivity index. This analyticalsolution may be executed on a computer system to quickly generateperformance parameters or other characteristics of the verticallyfractured well. This solution eliminates the need of performingtime-consuming numerical simulation for obtaining pseudo-steady statesolution for fractured wells in a near circular reservoir and it may beused to generate approximate solutions for reservoirs of othergeometrical shapes. For example, in comparison to the hours or daysrequired of a computer to generate solutions according to theconventional techniques described above, a computer may generatesolutions in accordance with described embodiments of the invention in amatter of seconds or minutes.

Described embodiments may yield a simple, exact expression for thepseudo-steady state constant b_(D,PSS), which can be used for variousapplications including production rate decline analysis and fracturedesign for optimized production. The solution can also be used as abenchmark to measure the accuracy of numerical simulations. Withsuitable shape factors, the analytical solution may be used to obtainapproximate expressions for the pseudo-steady state constant b_(D,PSS)for fractured wells in reservoirs of other geometrical shapes.

According to one embodiment, a method may include receiving a pluralityof shape factors corresponding to a geometrical shape of a hydraulicallyfractured well reservoir; determining a pseudo-steady state constant forthe reservoir based, at least in part, on the plurality of shapefactors; and/or determining a performance parameter of the reservoirwhen operated in a pseudo-steady state with a finite fractureconductivity based on the determined pseudo-steady state constant.

The foregoing has outlined rather broadly the features and technicaladvantages of the present invention in order that the detaileddescription of the invention that follows may be better understood.Additional features and advantages of the invention will be describedhereinafter that form the subject of the claims of the invention. Itshould be appreciated by those skilled in the art that the conceptionand specific embodiment disclosed may be readily utilized as a basis formodifying or designing other structures for carrying out the samepurposes of the present invention. It should also be realized by thoseskilled in the art that such equivalent constructions do not depart fromthe spirit and scope of the invention as set forth in the appendedclaims. The novel features that are believed to be characteristic of theinvention, both as to its organization and method of operation, togetherwith further objects and advantages will be better understood from thefollowing description when considered in connection with theaccompanying figures. It is to be expressly understood, however, thateach of the figures is provided for the purpose of illustration anddescription only and is not intended as a definition of the limits ofthe present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the disclosed system and methods,reference is now made to the following descriptions taken in conjunctionwith the accompanying drawings.

FIG. 1 are graphs illustrating reservoir pressure change with time for awell producing at constant rate in a closed reservoir according to theprior art.

FIG. 2 is a top view of a vertical well intersected by a thin ellipticalfracture according to some embodiments of the disclosure.

FIG. 3 is a flow chart illustrating an example method for computing ananalytical solution for pseudo-steady state flow for a verticallyfractured well with finite fracture conductivity in a closed reservoiraccording to some embodiments of the disclosure.

FIG. 4 are graphs of a pseudo-steady state constant computation as afunction of ξ_(e) calculated according to some embodiments of thedisclosure.

DETAILED DESCRIPTION

Consider fluid production from a fully-penetrated, vertically-fracturedwell from an initially quiescent state as shown in FIG. 2. FIG. 2 is atop view of a vertical well intersected by a thin elliptical fractureaccording to some embodiments of the disclosure. The drawing is forillustration purpose only and it does not reflect the actual scales. Insome embodiments, the fracture may be very thin and long, L>>w_(f), andthe fracture surface, ξ=μ₁≈0. The following commonly used assumptionsare made: the reservoir fluid is a single phase fluid residing in ahomogeneous medium with its motion governed by the Darcy's law in boththe reservoir and the fracture; the fluid and the reservoir are weaklycompressible, characterized by a single lumped total compressibilityconstant c₁; the effects of wellbore storage and skin are negligible;and the hydraulic fracture is supported by propants and it isincompressible. The hydraulic fracture is modeled as a thin, longellipse, intersecting the wellbore with a fracture width w_(f), which ismuch smaller than the wellbore diameter 2r_(w). The Cartesiancoordinates (x,y) and the elliptic coordinates (ξ,η) are related by x=Lcos hξcos η, y=L sin hξ sin η, with L being the focal distance which isessentially the fracture half-length; and L>>w_(f). The surface of thenarrow elliptical shape fracture is represented by the ellipse ξ=μ₁ inthe elliptic coordinates, and ξ₁ is a small number. Subscript “f” isused for reservoir and fracture quantities, respectively. Thepermeabilities in the reservoir and the hydraulic fracture are κ,κ_(f),respectively, with κ_(f)>>κ. For a successful hydraulic fracturing job,the fluid production is nearly entirely from the fracture, and thecontribution from the wellbore to the production is negligible. Thereservoir has a finite extent and its outer boundary is an ellipseξ=ξ_(e), confocal with the limiting ellipse ξ=ξ₁ used to represent thefracture. For mathematical simplicity, the finite drainage area isassumed to have an elliptical shape, which is a good geometricalapproximation to a large circular drainage area. A large circularreservoir with radius R can be well approximated as an ellipticalreservoir with ξ_(e)=ln(2R/L).

For the convenience of discussing the physical aspects of the analyticalsolution, we formulate the problem in terms of pressure instead ofpressure drawdown as in most of petroleum engineering literature.Pressure drawdown will be denoted as Δp throughout the paper. We choosethe reservoir initial pressure p_(i,d) and the pressure diffusion timescale as the characteristic pressure and characteristic time,respectively, for non-dimensionalization:p_(c)=p_(i,d),t_(c)=μφc_(t)L²/κ, where μ,φ are the fluid viscosity andreservoir porosity, respectively. The dimensionless reservoir pressuresatisfies a diffusion equation, which in an elliptical coordinates (ξ,η)becomes

$\begin{matrix}{{{\frac{\partial^{2}p_{D}}{\partial\xi^{2}} + \frac{\partial^{2}p_{D}}{\partial\eta^{2}}} = {\frac{{\cosh \; 2\; \xi} - {\cos \; 2\; \eta}}{2}\frac{\partial p_{D}}{\partial t_{DL}}}},} & (2)\end{matrix}$

where the dimensionless time and the dimensionless pressure are definedby

t _(DL) =κt/(μφc _(t) L ²),   (3)

p _(D) =p _(d) /p _(i,d).   (4)

Initially the reservoir fluid is at rest. Symmetry condition applies onthe x-axis and the y-axis; and the no-flow condition is imposed on thereservoir outer boundary,

$\begin{matrix}{\xi = {{\xi_{e}\text{:}\frac{\partial p_{D}}{\partial\xi}} = 0.}} & (5)\end{matrix}$

When the fracture compressibility is neglected, the dimensionlesspressure in the fracture, defined as

p _(fD)(η,t _(DL))=p _(f,d)(η,t _(DL))/p _(i,d),   (6)

satisfies the equation

$\begin{matrix}{{{{\frac{\partial^{2}{p_{fD}( {\eta,t_{DL}} )}}{\partial\eta^{2}} + {\frac{2}{F_{E}}\frac{\partial p_{D}}{\partial\xi}}}}_{\xi \; = \; \xi_{1}} = 0},} & (7)\end{matrix}$

where the dimensionless elliptical fracture conductivity

$\begin{matrix}{F_{E} = {\frac{\kappa_{f}w_{f}}{\kappa \; L}.}} & (8)\end{matrix}$

This elliptical fracture conductivity F_(E) is different from therectangular fracture conductivity commonly denoted as C_(fD). For anelliptical fracture, the width of the fracture is not a constant; F_(E)and C_(fD) only match with each other at the well. One way to relateF_(E) and C_(fD) is to assume that the elliptical fracture and therectangular fracture have the same volume, which leads toC_(fD)=πF_(E)/2. Symmetry condition on the x-axis is imposed and aconstant production rate at the wellbore is specified.

The Pseudo-Steady State Pressure Distribution in a Closed EllipticalReservoir

A pseudo-steady-state (PSS) solution is the long-time asymptoticsolution under constant production rate condition from a closedreservoir; and it has the property that

$\begin{matrix}{{\frac{\partial p_{D,{PSS}}}{\partial t_{DL}} = {{{const}.} = {{- C} < 0}}},} & (9)\end{matrix}$

where C is a dimensionless positive constant, C>0. Property (9) holdsfor any point in the reservoir. Thus, the reservoir pressure possessesthe form

p _(D,PSS)(ξ,η,t _(DL))={tilde over (p)}(ξ,η)−Ct _(DL),   (10)

and eqn. (2) becomes an eqn. for the shape function {tilde over(p)}(ξ,η):

$\begin{matrix}{{\frac{\partial^{2}\overset{\sim}{p}}{\partial\xi^{2}} + \frac{\partial^{2}\overset{\sim}{p}}{\partial\eta^{2}}} = {{- C}{\frac{{\cosh \; 2\; \xi} - {\cos \; 2\; \eta}}{2}.}}} & (11)\end{matrix}$

The solution to the inhomogeneous eqn. (11) can be written as

{tilde over (p)}(ξ,η)=p _(c)(ξ,η)+p _(p)(ξ,η),   (12)

where p_(c)(ξ,η) satisfies the homogeneous eqn.

$\begin{matrix}{{{\frac{\partial^{2}p_{c}}{\partial\xi^{2}} + \frac{\partial^{2}p_{c}}{\partial\eta^{2}}} = 0},} & (13)\end{matrix}$

and p_(p)(ξ,η) is a particular solution of the inhomogeneous eqn.

$\begin{matrix}{{\frac{\partial^{2}p_{p}}{\partial\xi^{2}} + \frac{\partial^{2}p_{p}}{\partial\eta^{2}}} = {{- C}{\frac{{\cosh \; 2\; \xi} - {\cos \; 2\; \eta}}{2}.}}} & (14)\end{matrix}$

One solution to equation (14) is

$\begin{matrix}{{p_{p}( {\xi,\eta} )} = {{- \frac{C}{8}}{( {{\cosh \; 2\xi} + {\cos \; 2\eta}} ).}}} & (15)\end{matrix}$

A possible solution to the homogeneous equation (13) has the form:

$\begin{matrix}{{{p_{C}( {\xi,\eta} )} = {B_{0} + {A_{0}\xi} - {\sum\limits_{n = 1}^{\infty}{A_{n}\cos \; 2n\; {\eta cosh2}\; {n( {\xi_{e} - \xi} )}}}}},} & (16)\end{matrix}$

where A_(i)(i=0,1,2 . . . ) are constants, and the symmetry conditionson the x-axis and y-axis (η=0,π/2) have already been satisfied. Theinfinite series enters the fracture eqn. because of its non-zero fluxdensity on the fracture surface. For the case of finite fractureconductivity, this infinite series is needed to match the non-constantfracture pressure inside the fracture.

Thus, the reservoir pressure for pseudo-steady state is given by

$\begin{matrix}{{p_{D,{PSS}}( {\xi,\eta,t_{DL}} )} = {B_{0} + {A_{0}\xi} - {\sum\limits_{n = 1}^{\infty}{A_{n}\cos \; 2n\; {\eta cosh2}\; {n( {\xi_{e} - \xi} )}}} - {\frac{C}{8}( {{\cosh \; 2\xi} + {\cos \; 2\eta}} )} - {{Ct}_{DL}.}}} & (17)\end{matrix}$

The no-flow outer boundary condition of equation (5) requires that

$\begin{matrix}{A_{0} = {{\frac{C}{4}\sinh \; 2\xi_{e}} > 0.}} & (18)\end{matrix}$

The complete pseudo-steady-state solution for the dimensionless pressurein the reservoir (ξ₁≦ξ≦ξ_(e)) is then

$\begin{matrix}{{p_{D,{PSS}}( {\xi,\eta,t_{DL}} )} = {B_{0} + {\frac{C}{4}{\xi sinh2\xi}_{e}} - {\sum\limits_{n = 1}^{\infty}{A_{n}\cos \; 2\; n\; {\eta cosh2}\; {n( {\xi_{e} - \xi} )}}} - {\frac{C}{8}( {{\cosh \; 2\xi} + {\cos \; 2\eta}} )} - {{Ct}_{DL}.}}} & (19)\end{matrix}$

The constant C is directly related to the fluid production rate. Thedimensional flux-density q_(d)(η) on the fracture surface ξ=ξ₁ for thefluid entering the fracture from the reservoir is given by

                                    (20) $\begin{matrix}{{{q_{d}(\eta)} = {\frac{\kappa}{\mu}\frac{p_{i,d}}{L\sqrt{{\sinh^{2}\xi_{1}} + {\sin^{2}\eta}}}\frac{\partial p_{D}}{\partial\xi}}}}_{\xi \; = \; \xi_{1}} \\{= {\frac{\kappa}{\mu}\frac{p_{i,d}}{L\sqrt{{\sinh^{2}\xi_{1}} + {\sin^{2}\eta}}}}} \\{{\lbrack {{\frac{C}{4}( {{\sinh \; 2\; \xi_{e}} - {\sinh \; 2\; \xi_{1}}} )} + {\sum\limits_{n = 1}^{\infty}{2n\; A_{n}\cos \; 2n\; \eta \; \sinh \; 2{n( {\xi_{e} - \xi_{1}} )}}}} \rbrack.}}\end{matrix}$

Therefore the dimensional production-rate for a bi-wing fractured-wellis

$\begin{matrix}{{Q_{d} = {{4h{\int_{0}^{\pi/2}{{q_{d}(\eta)}L\sqrt{{\sinh^{2}\xi_{1}} + {\sin^{2}\eta}}d\; \eta}}} = {\frac{\pi \; \kappa \; h\; p_{i,d}}{2\; \mu}{C( {{\sinh \; 2\; \xi_{e}} - {\sinh \; 2\; \xi_{1}}} )}}}},} & (21)\end{matrix}$

where h is the formation thickness. Thus, the dimensionless parameter Cis related to the well production rate by

$\begin{matrix}{C = {\frac{2\; \mu}{\pi \; \kappa \; h\; p_{i,d}}{\frac{Q_{d}}{{\sinh \; 2\; \xi_{e}} - {\sinh \; 2\; \xi_{1}}}.}}} & (22)\end{matrix}$

The Pseudo-Steady State Pressure Profile in the Fracture

The fracture pressure of equation (7) can be written as

$\begin{matrix}{{\frac{\partial^{2}{p_{{fD},{PSS}}( {\eta,t_{DL}} )}}{\partial\eta^{2}} + {\frac{2}{F_{E}}\lbrack {{\frac{C}{4}( {{\sinh \; 2\; \xi_{e}} - {\sinh \; 2\; \xi_{1}}} )} + {\sum\limits_{n = 1}^{\infty}{2n\; A_{n}\cos \; 2n\; \eta \; \sinh \; 2{n( {\xi_{e} - \xi_{1}} )}}}} \rbrack}} = 0.} & (23)\end{matrix}$

There is no-flow across the x-axis due to symmetry,

η=0: ∂p _(fD,PSS)/∂η=0,0≦ξ≦ξ₁.   (24)

The pseudo-steady state property also holds for the pressure inside thefracture,

$\begin{matrix}{\frac{\partial{p_{{fD},{PSS}}( {\eta,t_{DL}} )}}{\partial t_{DL}} = {- {C.}}} & (25)\end{matrix}$

Integration of equation (23) subject to equations (24) and (25) givesthe dimensionless fracture pressure

$\begin{matrix}{{{p_{{fD},{PSS}}( {\eta,t_{DL}} )} = {{- {\frac{2}{F_{E}}\lbrack {{\frac{C}{8}( {{\sinh \; 2\; \xi_{e}} - {\sinh \; 2\; \xi_{1}}} )\eta^{2}} - {\sum\limits_{n = 1}^{\infty}{\frac{A_{n}}{2n}\cos \; 2\; n\; \eta \; \sinh \; 2{n( {\xi_{e} - \xi_{1}} )}}}} \rbrack}} - {Ct}_{DL} + \overset{\sim}{C}}},} & (26)\end{matrix}$

where {tilde over (C)} is an integration constant.

The dimensionless pressure at the well, which is unknown for the PSSsolution, is given by

$\begin{matrix}{p_{{wD},{PSS}} = {{p_{{fD},{PSS}}( {{\pi \text{/}2},t_{DL}} )} = {{- {\frac{2}{F_{E}}\lbrack {{\frac{\pi^{2}C}{32}( {{\sinh \; 2\; \xi_{e}} - {\sinh \; 2\xi_{1}}} )} - {\sum\limits_{n = 1}^{\infty}{( {- 1} )^{n}\frac{A_{n}}{2n}\sinh \; 2{n( {\xi_{e} - \xi_{1}} )}}}} \rbrack}} - {Ct}_{DL} + {\overset{\sim}{C}.}}}} & (27)\end{matrix}$

Determination of the Coefficients

The coefficients A_(n) and the constant C in the solution for thepressure can be obtained by matching the reservoir pressure on thefracture surface with the fracture pressure and an application of thematerial balance equation. Because the fracture is narrow and ξ₁ is verysmall, we set ξ₁≈0 in all calculations below.

Pressure Matching on the Fracture Surface

On the fracture surface, ξ=ξ₁=0, the reservoir pressure and the fracturepressure must match,

p _(D,PSS)(0,η,t _(DL))=p _(fD,PSS)(η,t _(DL)).

This leads to equation (28):

$\begin{matrix}{{B_{0} - {\sum\limits_{n = 1}^{\infty}{A_{n}\cos \; 2n\; {\eta cosh2}\; n\; \xi_{e}}} - {\frac{C}{8}( {1 + {\cos \; 2\eta}} )} - {Ct}_{DL}} = {{- {\frac{2}{F_{E}}\lbrack {{\frac{C}{8}\eta^{2}\sinh \; 2\; \xi_{e}} - {\sum\limits_{n = 1}^{\infty}{\frac{A_{n}}{2n}\cos \; 2n\; {\eta sinh2}\; n\; \xi_{e}}}} \rbrack}} - {Ct}_{DL} + {\overset{\sim}{C}.}}} & (28)\end{matrix}$

It is observed that, without the infinite series in the reservoirpressure, it would not be possible to match the η² term from thefracture pressure. Pressure matching can be accomplished by simplyexpanding η² as a cosine series.

The Material Balance Equation

The dimensionless reservoir pressure drawdown Δp_(mD) is defined as

$\begin{matrix}{{\Delta \; p_{D}} = {\frac{2{\pi\kappa}\; h}{\mu \; Q_{d}} = {\lbrack {p_{i,d} - p_{d}} \rbrack = {{\frac{2{\pi\kappa}\; h\; p_{i,d}}{\mu \; Q_{d}}\lbrack {1 - p_{D}} \rbrack}.}}}} & (29)\end{matrix}$

For a closed reservoir and constant rate production, the materialbalance equation provides a simple relation between the reservoiraverage pressure drawdown and time,

$\begin{matrix}{{{\Delta \; {\overset{\_}{p}}_{D}} = {{\frac{2\pi \; \kappa \; h}{\mu \; Q_{d}}\lbrack {p_{i,d} - {p_{d}(t)}} \rbrack} = {2\pi \; t_{DA}}}},} & (30)\end{matrix}$

where p _(d)(t) is the reservoir volume-averaged pressure

$\begin{matrix}{{{{\overset{\_}{p}}_{d}(t)} = {\frac{1}{V}{\int_{V}{p_{d}\ {dv}}}}},} & (31)\end{matrix}$

V being the reservoir volume; and t_(DA) is the dimensionless timedefined in terms of the draining area, A=V/h=πL² sin h2ξ_(e)/2,

$\begin{matrix}{t_{DA} = {\frac{\kappa \; t}{\mu \; c_{t}\varphi \; A} = {\frac{2}{\pi}{\frac{t_{DL}}{{\pi sinh2\xi}_{e}}.}}}} & (32)\end{matrix}$

Computing the reservoir average pressure using the solution of equation(19) and utilizing the relation between the constant C and theproduction rate Q_(d) of equation (22), the material balance equation(30) becomes,

$\begin{matrix}{{{\int_{A}{\lbrack {1 - B_{0} - \ {\frac{C}{4}{\xi sinh2\xi}_{e}} + {\sum\limits_{n = 1}^{\infty}{A_{n}\cos \; 2n\; {\eta cosh2}\; {n( {\xi_{e} - \xi} )}}} + {\frac{C}{8}( {{\cosh \; 2\xi} + {\cos \; 2\eta}} )}} \rbrack {dA}}} = 0},\quad} & (33)\end{matrix}$

which leads to

$\begin{matrix}{B_{0} = {1 - {\frac{C}{4}( {{\xi_{e}\sinh \; 2\xi_{e}} - \frac{{\cosh \; 2\xi_{e}} - 1}{2}} )} + {\frac{C}{16}\cosh \; 2\xi_{e}} - {\frac{A_{1}}{2}.}}} & (34)\end{matrix}$

Matching Fourier coefficients in equation (28) then gives

$\begin{matrix}{{\overset{\sim}{C} = {1 - \frac{C}{4} - {\frac{C}{4}\xi_{e}\sinh \; 2\xi_{e}} + {\frac{3C}{16}\cosh \; 2\xi_{e}} - \frac{A_{1}}{2} + {\frac{C}{F_{E}}\frac{\pi^{2}}{48}\sinh \; 2\xi_{e}}}},} & (35) \\{\mspace{79mu} {{A_{1} = {- {\frac{C}{{\cosh \; 2\xi_{e}} + \frac{\sinh \; 2\xi_{e}}{F_{E}}}\lbrack {\frac{1}{8} + {\frac{1}{F_{E}}\frac{\sinh \; 2\xi_{e}}{4}}} \rbrack}}},}} & (36) \\{\mspace{79mu} {{A_{n} = {\frac{C}{4}\frac{( {- 1} )^{n}}{n}\frac{\sinh \; 2\xi_{e}}{{\sinh \; 2n\; \xi_{e}} + {n\; F_{E}\cosh \; 2n\; \xi_{e}}}}},{n \geq 2.}}} & (37)\end{matrix}$

Thus, the pressure in the reservoir and the fracture are completelydetermined. In particular, the dimensionless pressure drawdown in thereservoir is given by

$\begin{matrix}\begin{matrix}{{{\Delta \; {p_{D,{PSS}}( {\xi,\eta,t_{DL}} )}} = {{2\pi \; t_{DA}} + \xi_{e} - {\xi++}}}} \\{{{\frac{1}{\sinh \; 2\xi_{e}}\begin{bmatrix}{{- \frac{{3\cosh \; 2\xi_{e}} - 2}{4}} + {2a_{1}} +} \\{{4a_{1}\cos \; 2{{\eta cosh2}( {\xi_{e} - \xi} )}} +} \\\frac{{\cosh \; 2\xi} + {\cos \; 2\eta}}{2}\end{bmatrix}}, +}} \\{{\sum\limits_{n = 2}^{\infty}{\frac{( {- 1} )^{n}}{n}\frac{\cos \; 2n\; {\eta cosh}\; 2{n( {\xi_{e} - \xi} )}}{{\sinh \; 2n\; \xi_{e}} + {n\; F_{E}\cosh \; 2n\; \xi_{e}}}}}}\end{matrix} & (38) \\{\mspace{79mu} {where}} & \; \\{\mspace{79mu} {a_{1} = {\frac{A_{1}}{C} = {{- \frac{1}{8}}{{\frac{1}{{\cosh \; 2\xi_{e}} + \frac{\sinh \; 2\xi_{e}}{F_{E}}}\lbrack {1 + {\frac{2}{F_{E}}\sinh \; 2\xi_{e}}} \rbrack}.}}}}} & (39)\end{matrix}$

Shank's transformation can be used to accelerate the convergence of theinfinite series in equation (38).

The dimensionless pressure drawdown at the well is given by

$\begin{matrix}{{\Delta \; p_{{wD},{PSS}}} = {{2\; \pi \; t_{DA}} + \xi_{e} + \frac{1}{\sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} + \frac{2\; a_{1}}{\sinh \mspace{11mu} 2\xi_{e}} + {{\frac{1}{F_{E}}\lbrack {\frac{\pi^{2}}{6} + {4a_{1}} - {\sum\limits_{n = 2}^{\infty}\; {\frac{1}{n^{2}}\frac{1}{1 + {n\; F_{E}\coth \mspace{11mu} 2\; n\; \xi_{e}}}}}} \rbrack}.}}} & (40)\end{matrix}$

Thus, an explicit expression for the pseudo-steady state constantb_(D,PSS) is given by

$\begin{matrix}{b_{D,{PSS}} = {\xi_{e} + \frac{1}{\sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} + \frac{2\; a_{1}}{\sinh \mspace{11mu} 2\; \xi_{e}} + {{\frac{1}{F_{E}}\lbrack {\frac{\pi^{2}}{6} + {4a_{1}} - {\sum\limits_{n = 2}^{\infty}\; {\frac{1}{n^{2}}\frac{1}{1 + {n\; F_{E}\coth \mspace{11mu} 2\; n\; \xi_{e}}}}}} \rbrack}.}}} & (41)\end{matrix}$

In addition, the productivity index (PI) and the dimensionlessproductivity index (J_(D)) for the pseudo-steady state flow is given by

$\begin{matrix}{\begin{matrix}{J_{PSS} = {\frac{Q_{d}}{{\overset{\_}{P}}_{d} - P_{w,d}} = {\frac{\kappa \; h}{\mu}\frac{2\; \pi}{b_{D,{PSS}}}}}} \\{J_{D,{PSS}} = {{\frac{\mu}{2\; {\pi\kappa}\; h}J_{PSS}} = \frac{1}{b_{D,{PSS}}}}}\end{matrix}.} & (42)\end{matrix}$

The dimensionless productivity index J_(D), or the effective wellboreradius, can be used to characterize the productivity of unfractured andfractured wells. For example, J_(D,PSS) can be used for fracture design.

FIG. 3 is a flow chart illustrating an example method for computing ananalytical solution for pseudo-steady state flow for a verticallyfractured well with finite fracture conductivity in a closed reservoiraccording to some embodiments of the disclosure. A method 300 may beginat block 302 with receiving one or more shape factors corresponding to ageometrical shape of a hydraulically fractured well reservoir. The datareceived at block 302 may be received through, for example, an inputdevice or local storage coupled to a processor or may be receivedthrough a network communication from a remote data store or remote inputdevice. Examples of the one or more shape factors include ellipse focaldistance/fracture half-length, formation thickness, dimensionlesselliptical fracture conductivity, wellbore radius, radius of circulardrainage boundary, reservoir volume, fracture width at the wellbore,elliptical coordinates, elliptical fracture shape, and ellipticalreservoir shape.

Then, at block 304, a pseudo-steady state constant may be determined bythe processor, such as using equation (41) for the reservoir based onthe plurality of shape factors. The determination at block 304 may beperformed using one or more elementary functions to obtain an analyticalsolution and/or without solving Mathieu functions, which cansignificantly improve the computational speed of the determination incomparison to prior art numerical simulations. Block 304 mayalternatively or additionally include a computation of reservoirpressure drawdown from, for example, equation (38).

Next, at block 306, one or more performance parameters of the reservoirmay be determined by the processor when the reservoir is operated in apseudo-steady state with a finite fracture conductivity based on thedetermined pseudo-steady state constant. Although block 306 describesfinite fracture conductivity, infinite fracture conductivity mayalternatively be used for determining the performance parameter.Examples of performance parameters include a production decline rate fora reservoir, a total hydrocarbon reserves for a reservoir, aneconomically-recoverable reserves for a reservoir, the productivityindex (PI), and the dimensionless productivity index (J_(D)). Using theperformance parameters, decisions as to explore and produce from certainreservoirs may be made, and the improved information available from thepseudo-steady state analysis of the method 300 may increaseprofitability of the production from selected reservoirs. The one ormore performance parameters or the pseudo-steady state constant may bestored in local or remote storage, output to a display screen, orcommunicated to another device through a network communicationsconnection. Additional computations or decisions may be performed usingthe performance parameter, such as decisions relating to the productionof hydrocarbons from a particular reservoir.

The specific features of the method 300 for determining a pseudo-steadystate constant and a performance parameter from that constant results ina specific process for evaluating reservoirs using particularinformation and techniques. Analysis of reservoirs using the method 300results in a technological improvement over the prior art numericalsolutions, which are tedious simulations to process. The method 300 thusdescribes a process specifically designed to achieve an improvedtechnological result of decreased computational time and increasedcomputational accuracy in the conventional industry practice ofdetermining performance from reservoirs. Furthermore, the method 300,and particularly block 304, describes a new analytical solution forcalculation of parameters related to a reservoir that differs fromconventional industry solutions.

FIG. 4 are graphs of a pseudo-steady state constant computation as afunction of ξ_(e) calculated according to some embodiments of thedisclosure. The pseudo-steady state constant b_(D,PSS)(ξ_(e),F_(E)) isplotted against ξ_(e) for F_(E)=1,2,5,10,20,1000 in lines 402, 404, 406,408, 410, and 412 of FIG. 4, respectively. It is observed that for largeξ_(e), the slope ∂b_(D,PSS)/∂ξ_(e) becomes one, regardless of the valueof the fracture conductivity F_(E).

Comparison with Existing Results

The analytical solution obtained in the present work is exact andgeneral under the assumptions adopted, and the solution is valid forboth infinite and finite fracture conductivities. A comparison betweenthis new analytical solution and presently known results is providedbelow.

Infinite Fracture Conductivity.

One conventional pseudo-steady state solution for the case of infinitefracture conductivity, F_(E)→∞ shows that the dimensionless pressuredrawdown in the reservoir:

$\begin{matrix}{{{\Delta \; p_{D,{Prats}}} = {{2\; t_{D,{Prat}}} + \xi_{e} + \frac{1}{2\mspace{11mu} \sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} - \frac{1}{2\mspace{11mu} \sinh \mspace{11mu} 4\xi_{e}} - \xi - {\frac{1}{\sinh \mspace{11mu} 4\xi_{e}}\cosh \mspace{11mu} 2( {\xi_{e} - \xi} )\cos \mspace{11mu} 2\; \eta} + \frac{{\cosh \mspace{11mu} 2\xi} + {\cos \mspace{11mu} 2\eta}}{2\mspace{11mu} \sinh \mspace{11mu} 2\xi_{e}}}},} & (43)\end{matrix}$

where dimensionless time t_(D,Prat) is defined as related to t_(DA) by(after a correction to a missing factorφ in their definition):

$\begin{matrix}{t_{D,{Prats}} = {\frac{\pi \; \kappa \; t}{{\mu\varphi}\; c_{t}A} = {\pi \; {t_{DA}.}}}} & (44)\end{matrix}$

Thus, the dimensionless reservoir pressure drawdown from is:

$\begin{matrix}{{\Delta \; p_{D,{Prats}}} = {{2\pi \; t_{DA}} + \xi_{e} + \frac{1}{2\mspace{11mu} \sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} - \frac{1}{2\mspace{11mu} \sinh \mspace{11mu} 4\xi_{e}} - \xi - \frac{\cosh \mspace{11mu} 2( {\xi_{e} - \xi} )\cos \mspace{11mu} 2\; \eta}{\sinh \mspace{11mu} 4\xi_{e}} + {\frac{{\cosh \mspace{11mu} 2\xi} + {\cos \mspace{11mu} 2\eta}}{2\mspace{11mu} \sinh \mspace{11mu} 2\xi_{e}}.}}} & (45)\end{matrix}$

From equation (38), for infinite fracture conductivity, F_(E)→∞, theinfinite sum becomes zero, and

$a_{1} = {{- \frac{1}{8}}{\frac{1}{\cosh \mspace{11mu} 2\xi_{e}}.}}$

Thus, the reservoir dimensionless pressure drawdown becomes

${{\Delta \; {p_{D,{PSS}}( {\xi,\eta,t_{DL}} )}} = {{2\pi \; t_{DA}} + \xi_{e} - {{\xi++}{\frac{1}{\sinh \mspace{11mu} 2\xi_{e}}\begin{bmatrix}{{- \frac{{3\mspace{11mu} \cosh \mspace{11mu} 2\xi_{e}} - 2}{4}} - {\frac{1}{4}\frac{1}{\cosh \mspace{11mu} 2\xi_{e}}}} \\{\; {{{- \frac{1}{2}}\frac{1}{\cosh \mspace{11mu} 2\xi_{e}}\cos \mspace{11mu} 2\eta \mspace{11mu} \cosh \mspace{11mu} 2( {\xi_{e} - \xi} )} + \frac{{\cosh \mspace{11mu} 2\xi} + {\cos \mspace{11mu} 2\eta}}{2}}}\end{bmatrix}}}}},$

which is identical to the prior art result of equation (45).

Similarly, the pressure drawdown at the well from our solution becomes

$\begin{matrix}{{{\Delta \; p_{D,{PSS}}2\pi \; t_{DA}} + \xi_{e} - \frac{( {{3\mspace{11mu} \cosh \mspace{11mu} 2\xi} - 1} )( {{\cosh \mspace{11mu} 2\xi} - 1} )}{4\mspace{11mu} \sinh \mspace{11mu} 2\xi_{e}\mspace{11mu} \cosh \mspace{11mu} 2\xi_{e}}},} & (46)\end{matrix}$

which gives the pseudo-steady state constant for the case of infinitefracture conductivity

$\begin{matrix}{b_{D,{PSS}} = {\xi_{e} - {\frac{( {{3\mspace{11mu} \cosh \mspace{11mu} 2\xi_{e}} - 1} )( {{\cosh \mspace{11mu} 2\xi_{e}} - 1} )}{4\mspace{11mu} \sinh \mspace{11mu} 2\xi_{e}\mspace{11mu} \cosh \mspace{11mu} 2\xi_{e}}.}}} & (47)\end{matrix}$

In summary, in the limit of infinite fracture conductivity, ananalytical solution according to embodiments described herein matches aconventional solution for infinite fracture conductivity. Thisdemonstrates that the analytical model is correct, and that at least onespecific calculation matches a result from a conventional model.

Finite Fracture Conductivity.

For finite fracture conductivity, the pseudo-steady state constant alsodepends on the dimensionless fracture conductivityF_(E):b_(D,PSS)=b_(D,PSS)(ξ_(e),F_(E)). b_(D,PSS)(ξ_(e),F_(E)) has beencomputed in the prior art for selected sets of ξ_(e),F_(E) bysubtracting 2πt_(DA) from numerical simulation results for large times.This procedure involves numerical manipulation of the Mathieu functionsin the Laplace transform space as well as numerical inversion; and it istedious and time-consuming, as noted by these authors. Anonlinear-regression may be applied to fit such numerical results intoan empirical formula for b_(D,PSS)(ξ_(e),F_(E))

$\begin{matrix}{{{b_{D,{PSS}}( {\xi_{e},F_{E}} )} = {{1.00146\; \xi_{e}} + {0.0794849\; e^{- \xi_{e}}} - {0.16703\; u} + \frac{A}{B} - 0.754772}},} & (48)\end{matrix}$

with

u=ln F_(E),

A=a ₁ +a ₂ u+a ₃ u ² +a ₄ u ³ +a ₅ u ⁴ ,B=b ₁ +b ₂ u+b ₃ u ² +b ₄ u ³ +b₅ u ⁴,

a ₁=−4.7468,b ₁=−2.4941,

a ₂=36.2492,b ₂=21.6755,

a ₃=55.0998,b ₃=41.0303,

a ₄=−3.98311,b ₄=−10.4793,

a ₅=6.07102,b ₅=5.6108.

However, there are some apparent inconsistency and problematic issueswith equation (48): (i) the formula cannot re-produce certain tabulatedresults of the prior art; (ii) equation (48) can give rise to negativevalues of b_(D,PSS) when F_(E) becomes large; and it does not convergeto the exact result of the prior art for infinite fracture conductivity;(iii) when the empirical equation (48) is compared to the disclosedanalytical solution for b_(D,PSS)(ξ_(e),F_(E)) in equation (41), it isimmediately obvious that the coefficient for the linear term ξ_(e) inequation (48) must be “1.0”, instead of “1.00146.”

The results of the analytical solution of equation (41) are computed andcompared to corresponding values b_(D,PSS)(ξ_(e),F_(E)) for theparameter sets as known in the prior art. The results are shown below inour Table 1, where the results of prior art are listed in theparentheses for comparison. It is seen that the numerically computedvalues from the prior art generally agree very well with the describedanalytical solution.

TABLE 1 Values of b_(D,PSS)(ξ_(e),F_(E)) from the analytical solution.Values in the parentheses are those of prior art numerical simulations.ξ_(e) F_(E) = 1 F_(E) = 10 F_(E) = 100 F_(E) = 1000 0.25 0.849411(0.8481)  0.213087 (0.2150) 0.130127 (0.1306) 0.121565 (0.1220) 0.500.989853 (0.9902)  0.333336 (0.3337) 0.239246 (0.2396) 0.229383 (0.2298)0.75 1.16694 (1.1671) 0.460557 (0.4609) 0.353713 (0.3540) 0.342402(0.3426) 1.00  1.3632 (1.3627) 0.610541 (0.6109) 0.493289 (0.4936)0.480809 (0.4812) 1.25 1.57305 (1.5733) 0.787704 (0.7880) 0.663153(0.6634) 0.649857 (0.6501) 1.50 1.79635 (1.7963) 0.988308 (0.9884)0.858987 (0.8591) 0.845162 (0.8453) 1.75 2.02893 (2.0293)  1.20624(1.2067)  1.07391 (1.0743)  1.05975 (1.0602) 2.00 2.26787 (2.2682) 1.43597 (1.4363)  1.30178 (1.3021)  1.28741 (1.2877) 3.00 3.25252(3.2529)  2.40795 (2.4084)  2.27122 (2.2716)  2.25658 (2.2570) 4.004.25038 (4.2503)  3.40407 (3.4040)  3.26699 (3.2669)  3.25231 (3.2522)5.00 5.25009 (5.2486)  4.40354 (4.4021)  4.26642 (4.2649)  4.25173(4.2502)

Discussions

Analytical solutions for the reservoir pressure drawdown Δp_(D,PSS) andthe pseudo-steady state constant b_(D,PSS) are given by equations (38)and (41), respectively. These expressions may be exact forfully-penetrating hydraulically fractured vertical wells producing froma closed reservoir approximated as having an elliptical shape, and thesolutions are valid for both finite and infinite fractureconductivities. As a result of the analytical solution described inembodiments of the disclosure herein, tedious and time consumingnumerical simulations for obtaining pseudo-steady state solutions forfractured wells are no longer necessary for such reservoirs. For afractured-well in a large circular reservoir with a radius R, theseformulas can be readily applied with ξ_(e)=ln(2R/L), because a largecircle and an ellipse with large ξ_(e) are nearly identical. It is alsopossible to extend the expression for the pseudo-steady state constantb_(D,PSS) to a fractured-well in reservoirs of different geometricalshapes using an equivalent elliptical parameter ξ_(e) based on thereservoir drainage area or shape factors.

Conclusions

An exact analytical solution for pseudo-steady state productive flowfrom a fully-penetrating hydraulically fractured vertical well withfinite fracture conductivity in a closed reservoir approximated ashaving an elliptical shape is rigorously derived. The solution agreeswith prior art solutions in the limit of infinite fracture conductivity,and it agrees with the numerical results of the prior art for finitefracture conductivity. The analytical solution is exact, general andexpressed in terms of elementary functions; it is simple and easy toevaluate; and it completely eliminates the need of performing numericalsimulation for obtaining pseudo-steady state solution for a verticallyfractured well in such a reservoir. Simple expressions for thepseudo-steady state constant and the dimensionless productivity indexare described above. The solution may also be used to generateapproximate analytical solutions for pseudo-steady state flow from afractured-well in reservoirs of different geometrical shapes.

Advantages of Embodiments of the Invention

An exact analytical solution in the physical variable space forpseudo-steady state production from a vertically fractured well withfinite fracture conductivity in an elliptical reservoir is obtained fromthis work. The solution is expressed in terms of elementary functionsand it yields a simple, exact expression for the pseudo-steady stateconstant b_(D,PSS) and the dimensionless productivity index J_(D,PSS).This is the first time that an exact analytical solution has beenobtained for pseudo-steady state flow for a fractured well with finiteconductivity.

Some advantages resulting from this analytical solution are listedbelow:

(1) It eliminates the need to perform time-consuming numericalsimulation in order to obtain the pseudo-steady state constant b_(D,PSS)b_(D,PSS) and the dimensionless productivity index J_(D,PSS) forfractured wells in elliptical reservoirs, and it shortens the requiredcomputing time from hours/days to seconds;

(2) By introducing suitable shape factors, the solution can be used toobtain approximate expressions for the pseudo-steady state constantb_(D,PSS) for fractured wells in reservoirs of other geometrical shapes;

(3) The solution can be readily adopted for use with production declinemodels and simulators for estimating total hydrocarbon reserves in-placeas well as economically recoverable reserves;

(4) The solution can be used for optimal fracture design so that theproduction is optimized;

(5) The solution can be used as a benchmark to measure the accuracy ofvarious numerical simulators; and

(6) The techniques used in certain embodiments of the disclosure (suchas hyperbolic functions and Fourier series expansions) circumvent thecumbersome Mathieu functions commonly used in studying production fromfractured wells, and these techniques can be adopted for much wider usein studying similar problems.

Important of Pseudo Steady-State Flow Analysis

The duration of the pseudo-steady state flow and its productiveperformance largely determines the cumulative production of hydrocarbonform a well. The duration of pseudo-steady state flow is determined byhow fast the bottom-hole flowing pressure decreases to the lowestpermissible well pressure (critical pressure). Thus it is paramount toknow the change of the wellbore pressure with time. The pressuredrawdown (pressure drop from the initial reservoir pressure) at thewellbore is commonly expressed in dimensionless form as

Δp _(wD,PSS)=2πt _(D4) +b _(D,PSS),   (49)

where t_(DA) is the drainage area based dimensionless time, andb_(D,PSS) is the so-called pseudo-steady state constant which depends onthe reservoir model as well as the well/reservoir configuration. Thus,two parameters determine the duration of the pseudo-steady state flowperiod: the time-rate of decline, which is determined by the productionrate, and the pseudo-steady state constant.

The productive performance of a well is measured by the productivityindex, J, which is the amount of hydrocarbon produced per unit drop inthe reservoir average pressure. For pseudo-steady state flow, theproductivity index is inversely proportional to the pseudo-steady stateconstant b_(D,PSS),

$\begin{matrix}{{J = {\frac{\kappa \; h}{\mu}\frac{2\; \pi}{b_{D,{PSS}}}}},} & (50)\end{matrix}$

where κ,μ,h are the reservoir permeability, hydrocarbon viscosity, andhydrocarbon bearing formation thickness, respectively. Thus, theproductivity of a well during the pseudo-steady state period iscompletely determined by the pseudo-steady state constant b_(D,PSS).

Furthermore, pseudo-steady state solution has been often used in theproduction rate decline analysis because pseudo-steady state is the flowregime immediate preceding the production rate decline period (as shownin FIG. 1). Production rate decline can be used for estimating thehydrocarbon reserves in place and for assessing the economicallyrecoverable amount of hydrocarbon from a reservoir.

The pseudo-steady state flow analysis can be used to improve productionfrom reservoirs, because: Pseudo-steady state flow can impact thecumulative production of hydrocarbon from a well; the productiveperformance of a well can be assessed by evaluating the productivity ofthe well during the pseudo-steady state flow, which is determined by thevalue of the pseudo-steady state constant b_(D,PSS); Pseudo-steady stateflow can be used for estimating the total reserves in place in areservoir; and Pseudo-steady state flow can be used for estimating theeconomically recoverable amount of hydrocarbon from a reservoir.

Implementation

Computations described in the embodiments above may be executed on anysuitable processor-based device including, without limitation, personaldata assistants (PDAs), tablet computers, smartphones, computer gameconsoles, and multi-processor servers. Moreover, the systems and methodsof the present disclosure may be implemented on application specificintegrated circuits (ASIC), very large scale integrated (VLSI) circuits,or other circuitry.

If implemented in firmware and/or software, the functions describedabove may be stored as one or more instructions or code on acomputer-readable medium. Examples include non-transitorycomputer-readable media encoded with a data structure andcomputer-readable media encoded with a computer program.Computer-readable media includes physical computer storage media. Astorage medium may be any available medium that can be accessed by acomputer. By way of example, and not limitation, such computer-readablemedia can comprise RAM, ROM, EEPROM, CD-ROM or other optical diskstorage, magnetic disk storage or other magnetic storage devices, or anyother medium that can be used to store desired program code in the formof instructions or data structures and that can be accessed by acomputer. Disk and disc includes compact discs (CD), laser discs,optical discs, digital versatile discs (DVD), floppy disks and blu-raydiscs. Generally, disks reproduce data magnetically, and discs reproducedata optically. Combinations of the above should also be included withinthe scope of computer-readable media.

In addition to storage on computer readable medium, instructions and/ordata may be provided as signals on transmission media included in acommunication apparatus. For example, a communication apparatus mayinclude a transceiver having signals indicative of instructions anddata. The instructions and data are configured to cause one or moreprocessors to implement the functions outlined in the claims.

Although the present disclosure and its advantages have been describedin detail, it should be understood that various changes, substitutionsand alterations can be made herein without departing from the spirit andscope of the disclosure as defined by the appended claims. Moreover, thescope of the present application is not intended to be limited to theparticular embodiments of the process, machine, manufacture, compositionof matter, means, methods and steps described in the specification. Asone of ordinary skill in the art will readily appreciate from thepresent invention, disclosure, machines, manufacture, compositions ofmatter, means, methods, or steps, presently existing or later to bedeveloped that perform substantially the same function or achievesubstantially the same result as the corresponding embodiments describedherein may be utilized according to the present disclosure. Accordingly,the appended claims are intended to include within their scope suchprocesses, machines, manufacture, compositions of matter, means,methods, or steps.

What is claimed is:
 1. A method, comprising: receiving a plurality ofshape factors corresponding to a geometrical shape of a hydraulicallyfractured well reservoir; determining a pseudo-steady state constant forthe reservoir based, at least in part, on an analytical solutioninvolving the plurality of shape factors; and determining a performanceparameter of the reservoir when operated in a pseudo-steady state basedon the determined pseudo-steady state constant.
 2. The method of claim1, wherein the pseudo-steady state constant is computed according to thefollowing equation:$b_{D,{PSS}} = {\xi_{e} + \frac{1}{\sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} + \frac{2\; a_{1}}{\sinh \mspace{11mu} 2\; \xi_{e}} + {{\frac{1}{F_{E}}\lbrack {\frac{\pi^{2}}{6} + {4a_{1}} - {\sum\limits_{n = 2}^{\infty}\; {\frac{1}{n^{2}}\frac{1}{1 + {n\; F_{E}\coth \mspace{11mu} 2\; n\; \xi_{e}}}}}} \rbrack}.}}$3. The method of claim 1, wherein the step of determining thepseudo-steady state constant is based, at least in part, on one or moreelementary functions.
 4. The method of claim 1, wherein the step ofdetermining the pseudo-steady state constant is performed withoutsolving Mathieu functions.
 5. The method of claim 1, wherein the step ofdetermining a performance parameter comprises determining a productiondecline rate for the reservoir.
 6. The method of claim 1, wherein thestep of determining a performance parameter comprises determining atotal hydrocarbon reserves for the reservoir.
 7. The method of claim 1,wherein the step of determining a performance parameter comprisesdetermining economically recoverable reserves for the reservoir.
 8. Themethod of claim 1, wherein the geometrical shape is elliptical.
 9. Acomputer program product, comprising: a non-transitory computer readablemedium comprising code to execute the steps comprising: receiving aplurality of shape factors corresponding to a geometrical shape of ahydraulically fractured well reservoir; determining a pseudo-steadystate constant for the reservoir based, at least in part, on ananalytical solution involving the plurality of shape factors; anddetermining a performance parameter of the reservoir when operated in apseudo-steady state with a finite fracture conductivity based on thedetermined pseudo-steady state constant.
 10. The computer programproduct of claim 9, wherein the pseudo-steady state constant is computedaccording to the following equation:$b_{D,{PSS}} = {\xi_{e} + \frac{1}{\sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} + \frac{2\; a_{1}}{\sinh \mspace{11mu} 2\; \xi_{e}} + {{\frac{1}{F_{E}}\lbrack {\frac{\pi^{2}}{6} + {4a_{1}} - {\sum\limits_{n = 2}^{\infty}\; {\frac{1}{n^{2}}\frac{1}{1 + {n\; F_{E}\coth \mspace{11mu} 2\; n\; \xi_{e}}}}}} \rbrack}.}}$11. The computer program product of claim 9, wherein the step ofdetermining the pseudo-steady state constant is performed withoutsolving Mathieu functions.
 12. The computer program product of claim 9,wherein the step of determining a performance parameter comprisesdetermining a production decline rate for the reservoir.
 13. Thecomputer program product of claim 9, wherein the step of determining aperformance parameter comprises determining a total hydrocarbon reservesfor the reservoir.
 14. The computer program product of claim 9, whereinthe step of determining a performance parameter comprises determiningeconomically recoverable reserves for the reservoir.
 15. The computerprogram product of claim 9, wherein the geometrical shape is elliptical.16. An apparatus, comprising: a memory; and a processor coupled to thememory, wherein the processor is configured to execute the stepscomprising: receiving a plurality of shape factors corresponding to ageometrical shape of a hydraulically fractured well reservoir;determining a pseudo-steady state constant for the reservoir based, atleast in part, on an analytical solution involving the plurality ofshape factors; and determining a performance parameter of the reservoirwhen operated in a pseudo-steady state with a finite fractureconductivity based on the determined pseudo-steady state constant. 17.The apparatus of claim 16, wherein the pseudo-steady state constant iscomputed according to the following equation:$b_{D,{PSS}} = {\xi_{e} + \frac{1}{\sinh \mspace{11mu} 2\; \xi_{e}} - {\frac{3}{4}\coth \mspace{11mu} 2\; \xi_{e}} + \frac{2\; a_{1}}{\sinh \mspace{11mu} 2\; \xi_{e}} + {{\frac{1}{F_{E}}\lbrack {\frac{\pi^{2}}{6} + {4a_{1}} - {\sum\limits_{n = 2}^{\infty}\; {\frac{1}{n^{2}}\frac{1}{1 + {n\; F_{E}\coth \mspace{11mu} 2\; n\; \xi_{e}}}}}} \rbrack}.}}$18. The apparatus of claim 16, wherein the performance parametercomprises a production decline rate for the reservoir.
 19. The apparatusof claim 16, wherein the performance parameter comprises a totalhydrocarbon reserves for the reservoir.
 20. The apparatus of claim 16,wherein the performance parameter comprises economically recoverablereserves for the reservoir.